Does "cafe" relate to the word "cafeteria?"

Café

The origin and history of 'café' in English seems at first blush to be straightforward, well understood and well documented. For example, OED, in an entry not fully updated after its first publication in the 1888 edition, provides this account of the meaning of 'café' and its introduction into English from French:

A coffee-house, a restaurant; strictly a French term, but in the late 19th c. introduced into the English-speaking countries for the name of a class of restaurant. [Emphasis mine.]

Note the "late 19th" century attribution; presumably "late 19th c." means sometime before the 1888 date of first publication. (The previous, 1989 version of the entry uses the same wording; earlier versions of the entry are not available to me.)

OED follows with quotation evidence starting from 1802 (an earlier 1789 quotation is bracketed, indicating the English quotation includes café in French). The entry profile, however, suggests quotation evidence from 1763 ("first cited in Horace Walpole") was at one time part of the entry, but was corrected for the 1989 edition.

The 1802, 1815, and 1851 attestations of 'café' (not italicized) are likely cases OED considered "strictly" French. The 1870 attestation from D.J. Kirwan's Palace & Hovel, an account of an American's visit to London, most likely was not considered "strictly" French.

All that suggests 1870 was the date OED considers 'café' was "introduced" into English with the sense of "a class of restaurant". That date does not antedate the first OED attestation of 'cafeteria' in English, 1839. Nonetheless, the French term café could well have influenced the development of 'cafeteria'; the question is how and when, and whether other language influences were more prominent.

Cafeteria

'Cafeteria' has an oddly convoluted and difficult history in English. This history is not credited by OED's treatment, which starts by misdating what they provide as the first attestation, from 1839. Etymology Online perpetuates that misdating to 1839, I surmise, but does not provide the attestation for the 1839 date given, and so is not easily faulted.

cafeteria, n.
....
Etymology: < American-Spanish cafetería coffee-shop.
orig. U.S.
A coffee-house; a restaurant, esp. now a self-service restaurant.

1839 J. L. Stephens Trav. Russian & Turkish Emp. I. 157 Every third shop, almost, being a cafteria [sic] where a parcel of huge turbanded fellows were at their daily labours of smoking pipes and drinking coffee.

1894 Lakeside Directory Chicago 2188 Cafetiria Catering Co. 45 Lake.

OED

As mentioned, 1839 is not correct; the Stephens work was first serialized in 1836, in The American Monthly Magazine, v. 8. That serialization included several references to 'cafteria' (pp 482-4, 486), the context of which suggests these 'cafteria' were Turkish group lodgings where locals and travelers gathered to smoke, drink coffee, and sleep.

Additionally, but inconsequentially, the later publication of Stephens work in book form took place in 1838 (New York), rather than 1839. In that publication, 'cafteria' appears twice (pp 35, 156), and the context suggests a use in the sense of 'a class of restaurant', that is, the same sense accorded to the later, 1870, English term 'café'.

Antedating 'Cafeteria'

The OED attestation from 1839, and the slightly earlier appearances of 'cafeteria' in works by the same author are not, however, the earliest appearances of the term in English. Earlier appearances suggest the word entered English not from American-Spanish, but rather from Spanish or Italian (Castel Nuovo is in Naples, Italy).

In The Eclectic Review of 1833, this mention is made:

...the Spaniard's lemonade is tasteless unless he can sip it in the accustomed Caffeteria...

And in The London Literary Gazette of 1829, in an exerpt from an 1828 work (Reisen in Egypten, &c. Travels in Egypt, Libya, Nubia, and Dongola, Between the Years 1820-1825, W.F. Hemprich and C.G. Ehrenberg), this:

...nor can I ever forget the five living Rolandos whom I saw standing in the caffeteria of Castel Nuovo...

Although neither of the attestations provides enough context to reliably ascertain the nature of the Spanish or Italian 'caffeteria', it seems clear that such were, generally, coffeehouses or "a class of" restaurants.

These attestations demonstrate that 'caffeteria' was in use in British English, with a sense at least similar to 'café', before it was adopted from American-Spanish into US English and began its progress toward the contemporary sense of 'self-service restaurant'.

Another Consideration

Noting the 55 (or more accurately, 58) year gap between the first and second OED attestations, I made shift to fill it. This was not so easily accomplished.

In an 1870 work, Italy: Handbook for Travellers, by K. Bædeker, I found this, where 'Caffeteria' is obviously Italian, in the section listing establishments of interest to tourists in Bologna:

Cafés. ...Beer: Birraria e Caffeteria della Ditta Neviani in the side passage of the Piazza, E. of S. Petronio...

Cafeteria in US English

I found no uses of 'cafeteria' (or variants) in British or US English between 1839 and 1894. Numerous uses of 'cafetiere', meaning coffeepot, do appear, but do not seem on point for this answer.

Looking at the 1894 uses of 'cafeteria' did provide some interesting background, although the credibility of the claims made by the complainant in this March, 1894 newspaper article (The Inter Ocean, Chicago, Illinois, p 10; paywalled) has been undermined by the foregoing attestations of 'caffeteria' in English, possibly Spanish, and Italian:

John Kroger, the proprietor of the "Cafeteria" Catering Company, has filed a bill in the Circuit Court to restrain W. H. Dittmer from using the name "Cafetier" in connection with the latter's restaurant. The bill says that the complainant adopted the name "Cafeteria" as a fanciful and arbitrary name for his business, although he says that the word has no descriptive meaning, and is not found in any language.

Also of passing interest is this April, 1894 (The Worthington Advance, Worthington, Minnesota, p 6; paywalled) description, which makes plain that the sense of 'cafeteria' as "a self-service restaurant" has been established:

These new places are called cafetiria, or cafetier, or cafelakota, or buffetiria, or some other meaningless variation on a common name, and their motto is every man is his own waiter.

Brazil exports coffee to England since the XVI century, maybe some portuguese words can be helpful to understand the question:

Cafezal: Coffee plantation (where farmers work)

Café da manhã: breakfast (which includes café)

Cafeteria: place where the workers eat, including "café da manhã"

Cafeteira: coffee jar

As a Portuguese speaker, all these words seem totally related.

What seems to be missing from all the other answers is the meaning of the suffix -eria.

In Italian, the -eria suffix can mean "a place connected with". So a pizzeria sells pizza and a gelateria sells gelato.

In Spanish, the -ería suffix means "a place that sells something." So a heladería is an ice cream parlor and a florería is a florist.

So cafetería (in Spanish) etymologically means a place you could buy and drink coffee. I don't know why there's an extra "t" in it; that is, why isn't it simply cafería. It may be because cafetera means coffee-pot in Spanish. Wiktionary thinks the extra "t" in cafetera was influenced by the "t" in tetera (teapot). And presumably there's an extra "t" in tetera because you te is a one-syllable word in Spanish, so you need to add a consonant when you add the suffix -era.